Let $R$ be a commutative ring. Could anyone advise me on how to prove $R$ has $\text{ACCP}$ (Ascending chain condition for principal ideals) iff every collection of principal ideals of $R$ has maximal element?
Hints will suffice, thank you.
2026-04-13 16:10:58.1776096658
A problem on $\text{ACCP}$
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It is easy to see how the second implies the first. Ascending chains are certainly collections, so if they always have maximal elements then ACCP holds.
The first implies the second by definition of maximal element. Suppose towards a contradiction that R satisfies ACCP yet there is some collection of principal ideals with no maximal element. Then any ideal in this collection is not maximal. Thus there is another ideal in the collection which contains it, and so on. Which would constitute an infinite strictly increasing chain of principal ideals.